Finite Group Databasebeta

← C2×#12796 #493696

Subgroups — C2×#12796 #493696

Order 512.

1400 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2322C211yesmaximal, characteristic
2C22×#352C212yesmaximal
3C2×#1800C211yesmaximal, characteristic
4C2×#1799C211yesmaximal, characteristic
5#12796C218yesmaximal, direct product factor
6C2×#1390C212yesmaximal
7C22×#123C2211yescharacteristic
8C22×C4×D4C2211yescharacteristic
9C23×C4⋊C4C411yescharacteristic, complemented
10C2×#423—21no—
11C22×#100C2212yes—
12C22×#99C2212yes—
13#2322—22no—
14#2322—22no—
15#2322C414yescomplemented
16#2322C2214yes—
17C2×#389—21no—
18C2×#388—21no—
19C2×#387—21no—
20C2×#386C2211yescharacteristic
21C2×#385—21no—
22C2×#385C411yescharacteristic, complemented
23C2×#381C411yescharacteristic, complemented
24C2×#380—21no—
25C2×C4×C22⋊C4C411yescharacteristic, complemented
26C2×#352—28no—
27C2×#352C2218yes—
28#1800—22no—
29#1800C2214yes—
30#1799—22no—
31#1799C2214yes—
32C2×#343—21no—
33C2×#342—21no—
34#1390C2218yes—
35C23×D4C2311yescharacteristic
36C24×C4C2×C411yescharacteristic
37C24×C4C2×C411yescharacteristic
38C24×C4C2×C411yescharacteristic
39C24×C4D411yescharacteristic
40C22×M16—22no—
41C2×#123—21no—
42C2×#123—21no—
43C2×#123—22no—
44C2×#123—22no—
45C2×#123C2×C412yes—
46C2×#123D414yes—
47C2×#123Q814yes—
48C2×#123C2312yes—
49C2×C4×D4—41no—
50C2×C4×D4—41no—
51C2×C4×D4—42no—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×C4×D4C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—42no—
62C22×C4⋊C4—41no—
63C22×C4⋊C4—41no—
64C22×C4⋊C4—41no—
65C22×C4⋊C4C2×C414yescomplemented
66C22×C4⋊C4D411yescharacteristic
67C22×C22⋊C4—21no—
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4C2×C411yescharacteristic
71C22×C22⋊C4C2×C411yescharacteristic
72C22×C22⋊C4C2×C411yescharacteristic
73C22×C22⋊C4D411yescharacteristic
74C22×C42—21no—
75C22×C42D411yescharacteristic
76#423—24no—
77#423—42no—
78C2×C4⋊C8—41no—
79C2×C4⋊C8—41no—
80C2×#100—24no—
81C2×#100—24no—
82C2×#100—22no—
83C2×#100C2314yes—
84C2×#99—24no—
85C2×#99—24no—
86C2×#99—22no—
87C2×#99C2314yes—
88#389—24no—
89#389—24no—
90#388—24no—
91#388—24no—
92#387—24no—
93#387—24no—
94#386—21no—
95#386—21no—
96#386C2×C412yes—
97#386C2312yes—
98#385—24no—
99#385—24no—
100#385—22no—
101#385C2×C414yescomplemented
102#381—22no—
103#381C2×C414yescomplemented
104#380—24no—
105#380—42no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4C2×C414yescomplemented
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—41no—
114#352—216no—
115#343—24no—
116#342—24no—
117C25C22×C411yescharacteristic
118C25C2×D411yescharacteristic
119C22×D4—41no—
120C22×D4—41no—
121C22×D4—42no—
122C22×D4C22×C411yescharacteristic
123C22×D4C2×D411yescharacteristic
124C22×D4C2×D412yes—
125C22×D4C2×D411yescharacteristic
126C22×D4C2×Q812yes—
127C22×D4#4112yes—
128C22×D4#4112yes—
129C22×D4C2411yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—41no—
144C23×C4—41no—
145C23×C4—42no—
146C23×C4—41no—
147C23×C4—41no—
148C23×C4—42no—
149C23×C4—82no—
150C23×C4—82no—
151C23×C4—82no—
152C23×C4C22⋊C411yescharacteristic
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C412yes—
156C23×C4C22⋊C412yes—
157C23×C4C22⋊C411yescharacteristic
158C23×C4C4⋊C414yescomplemented
159C23×C4C22×C412yes—
160C23×C4C22×C412yes—
161C23×C4C22×C412yes—
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C2×M16—24no—
165C2×M16—42no—
166C2×M16—44no—
167C2×M16—44no—
168C22×C8—42no—
169#123—24no—
170#123—24no—
171#123—24no—
172#123—24no—
173#123—24no—
174#123—22no—
175#123—22no—
176#123C4⋊C418yescomplemented
177#123C2×D414yes—
178#123C2×Q814yes—
179C4×D4—42no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183C4×D4—42no—
184C4×D4—42no—
185C4×D4—42no—
186C4×D4—42no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4C22⋊C412yes—
209C2×C4⋊C4C2×D412yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—24no—
214C2×C22⋊C4—24no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—42no—
225C2×C22⋊C4—44no—
226C2×C22⋊C4—44no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—82no—
229C2×C22⋊C4C22⋊C412yes—
230C2×C22⋊C4C22⋊C412yes—
231C2×C22⋊C4C4⋊C414yescomplemented
232C2×C22⋊C4C22×C412yes—
233C2×C22⋊C4C22×C412yes—
234C2×C22⋊C4C22×C412yes—
235C2×C22⋊C4C2×D412yes—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—82no—
241C2×C42C22⋊C412yes—
242C2×C42C2×D412yes—
243C4⋊C8—44no—
244C4⋊C8—44no—
245#100—24no—
246#100—24no—
247#100—44no—
248#99—24no—
249#99—24no—
250#99—44no—
251#95—24no—
252#95—24no—
253#95—24no—
254#95—24no—
255#95—22no—
256#95—22no—
257#95—44no—
258C24—21no—
259C24—21no—
260C24—21no—
261C24—22no—
262C24—21no—
263C24—22no—
264C24—42no—
265C24—42no—
266C24—42no—
267C24—42no—
268C24C2×C22⋊C411yescharacteristic
269C24C2×C22⋊C411yescharacteristic
270C24C2×C22⋊C411yescharacteristic
271C24C2×C4⋊C412yes—
272C24#11712yes—
273C24#12112yes—
274C24C22⋊Q812yes—
275C24C23×C411yescharacteristic
276C24C22×D411yescharacteristic
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—41no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—42no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4C2×C4⋊C412yes—
294C2×D4#11712yes—
295C2×D4#12112yes—
296C2×D4#12112yes—
297C2×D4C22⋊Q812yes—
298C2×D4C22⋊Q812yes—
299C2×D4C22×D411yescharacteristic
300C2×D4C22×Q811yescharacteristic
301C2×D4C2×#4111yescharacteristic
302C2×D4C2×#4111yescharacteristic
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—42no—
329C22×C4—44no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—42no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—44no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4—82no—
360C22×C4—82no—
361C22×C4—84no—
362C22×C4—84no—
363C22×C4—82no—
364C22×C4—82no—
365C22×C4—82no—
366C22×C4—82no—
367C22×C4—84no—
368C22×C4—84no—
369C22×C4—84no—
370C22×C4—84no—
371C22×C4—84no—
372C22×C4#9911yescharacteristic
373C22×C4#9911yescharacteristic
374C22×C4#9912yes—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C412yes—
377C22×C4C2×C22⋊C412yes—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C4⋊C414yescomplemented
381C22×C4#12112yes—
382C22×C4C22⋊Q812yes—
383C22×C4C22×D411yescharacteristic
384M16—44no—
385M16—44no—
386M16—84no—
387C2×C8—44no—
388C2×C8—84no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—28no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—48no—
395C4⋊C4—44no—
396C4⋊C4—44no—
397C4⋊C4—84no—
398C4⋊C4—88no—
399C22⋊C4—22no—
400C22⋊C4—24no—
401C22⋊C4—24no—
402C22⋊C4—28no—
403C22⋊C4—24no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—44no—
411C22⋊C4—44no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—84no—
415C22⋊C4C2×C4⋊C414yescomplemented
416C42—22no—
417C42—22no—
418C42—22no—
419C42—22no—
420C42—84no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—22no—
429C23—21no—
430C23—21no—
431C23—21no—
432C23—21no—
433C23—21no—
434C23—21no—
435C23—22no—
436C23—22no—
437C23—21no—
438C23—21no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23—44no—
443C23—44no—
444C23—41no—
445C23—42no—
446C23—42no—
447C23—48no—
448C23—41no—
449C23—41no—
450C23—41no—
451C23—44no—
452C23—44no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—42no—
457C23—42no—
458C23—81no—
459C23—81no—
460C23—84no—
461C23#38012yes—
462C23#38012yes—
463C23#38012yes—
464C23C2×#9911yescharacteristic
465C23C2×#9911yescharacteristic
466C23#41012yes—
467C23C22×C22⋊C411yescharacteristic
468C23C22×C4⋊C411yescharacteristic
469C23C2×#11711yescharacteristic
470C23C2×#12111yescharacteristic
471C23C2×C22⋊Q811yescharacteristic
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—41no—
479D4—41no—
480D4—41no—
481D4—41no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—24no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—44no—
500C2×C4—44no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—48no—
504C2×C4—44no—
505C2×C4—48no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—44no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—82no—
527C2×C4—82no—
528C2×C4—82no—
529C2×C4—82no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—84no—
533C2×C4—82no—
534C2×C4—84no—
535C2×C4—82no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—168no—
541C2×C4#38012yes—
542C2×C4C2×#9912yes—
543C2×C4C2×#9912yes—
544C2×C4C2×#12111yescharacteristic
545C2×C4C2×C22⋊Q811yescharacteristic
546C8—84no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—24no—
552C22—24no—
553C22—21no—
554C22—21no—
555C22—22no—
556C22—21no—
557C22—21no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—44no—
568C22—44no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—44no—
578C22—44no—
579C22—44no—
580C22—44no—
581C22—81no—
582C22—81no—
583C22—82no—
584C22—82no—
585C22#179912yes—
586C22#179912yes—
587C22#212112yes—
588C22#212312yes—
589C22C2×#38011yescharacteristic
590C22C22×#9911yescharacteristic
591C22C2×#41011yescharacteristic
592C4—22no—
593C4—22no—
594C4—21no—
595C4—21no—
596C4—22no—
597C4—44no—
598C4—42no—
599C4—42no—
600C4—42no—
601C4—44no—
602C4—82no—
603C4—82no—
604C4—164no—
605C2—21no—
606C2—21no—
607C2—22no—
608C2—41no—
609C2—41no—
610C2—41no—
611C2—42no—
612C2—41no—
613C2—42no—
614C2#1279614yesdirect product factor
615C2C2×#179911yescharacteristic
616C2C2×#212111yescharacteristic
617C2C2×#212311yescharacteristic
subgroupG/H
center, socleC23C2×#99
commutatorC24C23×C4
Frattini, cosocleC22×D4C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4×D4 ◁ G
LECSG ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #12796C2direct (N × H)
9. C23×C4⋊C4C4semidirect
15. #2322C4semidirect
22. C2×#385C4semidirect
23. C2×#381C4semidirect
25. C2×C4×C22⋊C4C4semidirect
65. C22×C4⋊C4C2×C4semidirect
101. #385C2×C4semidirect
103. #381C2×C4semidirect
107. C4×C22⋊C4C2×C4semidirect
158. C23×C4C4⋊C4semidirect
176. #123C4⋊C4semidirect
231. C2×C22⋊C4C4⋊C4semidirect
380. C22×C4C2×C4⋊C4semidirect
415. C22⋊C4C2×C4⋊C4semidirect
614. C2#12796direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#23222
C22×#3522
C2×#18002
C2×#17992
#127962
C2×#13902